[Here][1] is the case $\alpha=1/2.$ with rounding down. It shows conjectures but few results. But you asked about rounding up. That is [here][2] and it seems that more is known in this $\alpha=1/2$ case including a good estimte of the growth rate. The $nth$ term is the number of partitions of $2n$ into powers of 2.


  [1]: http://oeis.org/A033485
  [2]: http://eis.org/A000123